# Map-State DualityΒΆ

```>>> from qitensor import qubit, qudit
>>> # input spaces for bipartite unitary
>>> hA = qudit('A', 3)
>>> hB = qudit('B', 4)
>>> # output spaces for bipartite unitary
>>> hAbar = qudit('Abar', 2)
>>> hBbar = qudit('Bbar', 6)
>>> # create a unitary
>>> U = (hAbar*hBbar*hA.H*hB.H).random_unitary()
>>> U.space
|Abar,Bbar><A,B|
```
```>>> # the cross operator (partial transpose)
>>> Ux = U.relabel({ hA.H: hA, hBbar: hBbar.H })
>>> Ux.space
|A,Abar><B,Bbar|
>>> # do a Schmidt decomposition
>>> hc = qudit('c', 6)
>>> (VA, M, VB) = Ux.svd(inner_space=hc, full_matrices=False)
>>> VA.space
|A,Abar><c|
>>> M.space
|c><c|
>>> VB.space
|c><B,Bbar|
>>> Ux.closeto(VA*M*VB)
True
```
```>>> # VA is an isometry, so it is the channel ket of a CPTP map.
>>> # Here is the superoperator for this map:
>>> E = lambda x: (VA * x * VA.H).trace(hAbar)
```
```>>> # Similarly, construct a channel for HB, but this will act on space c'
>>> VBprime = VB.relabel({ hc: hc.prime })
>>> VBprime.space
|c'><B,Bbar|
>>> # How about transposing it...
>>> VBprime = VBprime.T
>>> VBprime.space
|B,Bbar><c'|
>>> # the superoperator:
>>> F = lambda x: (VBprime * x * VBprime.H).trace(hBbar)
```
```>>> # The M matrix can be converted to a state by doing a partial
>>> # transpose.  It is not allowed to have the same space repeated
>>> # twice, so prime one of them.
>>> psi = M.relabel({ hc.H: hc.prime }).normalized()
>>> psi.space
|c,c'>
>>> # make a density operator
>>> rho = psi.O # shortcut for writing psi*psi.H
>>> rho.space
|c,c'><c,c'|
>>> # pass this through the channels
>>> E(rho).space
|A,c'><A,c'|
>>> sigma = F(E(rho))
>>> sigma.space
|A,B><A,B|
>>> # The result is the fully mixed state on |A,B>.  Can you see why?
>>> sigma.closeto( (hA*hB).eye()/12 )
True
```

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